Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping

This work is concerned with a semilinear non-homogeneous Timoshenko system under the effect of two nonlinear localized frictional damping mechanisms. The main goal is to prove its uniform stability by imposing minimal amount of support for the damping and, as expected, without assuming any relation on the non-constant coefficients. This fact generalizes substantially the previous papers by Cavalcanti et al. (Z Angew Math Phys 65(6):1189–1206, 2014) and Santos et al. (Differ Integral Equ 27(1–2):1–26, 2014) at the levels of problem and method. It is worth mentioning that the methodologies of these latter cannot be applied to the semilinear case herein, namely when one considers the problem with nonlinear source terms. Thus, differently of Cavalcanti et al. (Z Angew Math Phys 65(6):1189–1206, 2014), Santos et al. (Differ Integral Equ 27(1–2):1–26, 2014), the proof of our main stability result relies on refined arguments of microlocal analysis due to Burq and Gérard (Contrôle Optimal des équations aux dérivées partielles, http://www.math.u-psud.fr/~burq/articles/coursX.pdf, 2001). As far as we know, it seems to be the first time that such a methodology has been employed to 1-D systems of Timoshenko type with nonlinear foundations.